1. Functions and Their Graphs Chapter 2 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AAA A

2. Functions Section 2.1

3. Relations Relation: A correspondence between two sets. x corresponds to y or y depends on x if a relation exists between x and y Denote by x ! y in this case.

4. Relations Example. Melissa John Jennifer Patrick $45,000 $40,000 $50,000 PersonSalary

5. Relations Example. 014014 0 1 {1 2 {2 Number

6. Functions Function: special kind of relation Each input corresponds to precisely one output If X and Y are nonempty sets, a function from X into Y is a relation that associates with each element of X exactly one element of Y

7. Functions Example. Problem: Does this relation represent a function? Answer: Melissa John Jennifer Patrick $45,000 $40,000 $50,000 PersonSalary

8. Functions Example. Problem: Does this relation represent a function? Answer: 014014 0 1 {1 2 {2 Number

9. Domain and Range Function from X to Y Domain of the function: the set X. If x in X: The image of x or the value of the function at x: The element y corresponding to x Range of the function: the set of all values of the function

10. Domain and Range Example. Problem: What is the range of this function? Answer: 01490149 {3 {2 {1 0 1 2 3 XY

11. Domain and Range Example. Determine whether the relation represents a function. If it is a function, state the domain and range. Problem: Relation: f (2,5), (6,3), (8,2), (4,3) g Answer:

12. Domain and Range Example. Determine whether the relation represents a function. If it is a function, state the domain and range. Problem: Relation: f (1,7), (0, {3), (2,4), (1,8) g Answer:

13. Equations as Functions To determine whether an equation is a function Solve the equation for y. If any value of x in the domain corresponds to more than one y, the equation doesn ’ t define a function Otherwise, it does define a function.

14. Equations as Functions Example. Problem: Determine if the equation x + y 2 = 9 defines y as a function of x. Answer:

15. Function as a Machine Accepts numbers from domain as input. Exactly one output for each input.

16. Finding Values of a Function Example. Evaluate each of the following for the function f(x) = {3x 2 + 2x (a) Problem: f(3) Answer: (b) Problem: f(x) + f(3) Answer: (c) Problem: f({x) Answer: (d) Problem: {f(x) Answer: (e) Problem: f(x+3) Answer:

17. Finding Values of a Function Example. Evaluate the difference quotient of the function Problem: f(x) = { 3x 2 + 2x. Answer:

18. Implicit Form of a Function A function given in terms of x and y is given implicitly. If we can solve an equation for y in terms of x, the function is given explicitly

19. Implicit Form of a Function Example. Find the explicit form of the implicit function. (a) Problem: 3x + y = 5 Answer: (b) Problem: xy + x = 1 Answer:

20. Important Facts For each x in the domain of f, there is exactly one image f(x) in the range An element in the range can result from more than one x in the domain We usually call x the independent variable y is the dependent variable

21. Finding the Domain If the domain isn ’ t specified, it will always be the largest set of real numbers for which f(x) is a real number We can ’ t take square roots of negative numbers (yet) or divide by zero

22. Finding the Domain Example. Find the domain of each of the following functions. (a) Problem: f(x) = x 2 { 9 Answer: (b) Problem: Answer: (c) Problem: Answer:

23. Finding the Domain Example. A rectangular garden has a perimeter of 100 feet. (a) Problem: Express the area A of the garden as a function of the width w. Answer: (b) Problem: Find the domain of A(w) Answer:

24. Operations on Functions Arithmetic on functions f and g Sum of functions: (f + g)(x) = f(x) + g(x) Difference of functions: (f { g)(x) = f(x) { g(x) Domains: Set of all real numbers in the domains of both f and g. For both sum and difference

25. Operations on Functions Arithmetic on functions f and g Product of functions f and g is (f ¢ g)(x) = f(x) ¢ g(x) The quotient of functions f and g is Domain of product: Set of all real numbers in the domains of both f and g Domain of quotient: Set of all real numbers in the domains of both f and g with g(x)  0

26. Operations on Functions Example. Given f(x) = 2x 2 + 3 and g(x) = 4x 3 + 1. (a) Problem: Find f+g and its domain Answer: (b) Problem: Find f { g and its domain Answer:

27. Operations on Functions Example. Given f(x) = 2x 2 + 3 and g(x) = 4x 3 + 1. (c) Problem: Find f ¢ g and its domain Answer: (d) Problem: Find f/g and its domain Answer:

28. Key Points Relations Functions Domain and Range Equations as Functions Function as a Machine Finding Values of a Function Implicit Form of a Function Important Facts Finding the Domain

29. Key Points (cont.) Operations on Functions

30. The Graph of a Function Section 2.2

31. Vertical-line Test Theorem. [Vertical-Line Test] A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graphs in at most one point.

32. Vertical-line Test Example. Problem: Is the graph that of a function? Answer:

33. Vertical-line Test Example. Problem: Is the graph that of a function? Answer:

34. Finding Information From Graphs Example. Answer the questions about the graph. (a) Problem: Find f(0) Answer: (b) Problem: Find f(2) Answer: (c) Problem: Find the domain Answer: (d) Problem: Find the range Answer:

35. Finding Information From Graphs Example. Answer the questions about the graph. (e) Problem: Find the x-intercepts: Answer: (f) Problem: Find the y-intercepts: Answer:

36. Finding Information From Graphs Example. Answer the questions about the graph. (g) Problem: How often does the line y = 3 intersect the graph? Answer: (h) Problem: For what values of x does f(x) = 2? Answer: (i) Problem: For what values of x is f(x) > 0? Answer:

37. Finding Information From Formulas Example. Answer the following questions for the function f(x) = 2x 2 { 5 (a) Problem: Is the point (2,3) on the graph of y = f(x)? Answer: (b) Problem: If x = {1, what is f(x)? What is the corresponding point on the graph? Answer: (c) Problem: If f(x) = 1, what is x? What is (are) the corresponding point(s) on the graph? Answer:

38. Key Points Vertical-line Test Finding Information From Graphs Finding Information From Formulas

39. Properties of Functions Section 2.3

40. Even and Odd Functions Even function: For every number x in its domain, the number { x is also in the domain f( { x) = f(x) Odd function: For every number x in its domain, the number { x is also in the domain f( { x) = { f(x)

41. Description of Even and Odd Functions Even functions: If (x, y) is on the graph, so is ( { x, y) Odd functions: If (x, y) is on the graph, so is ( { x, { y)

42. Description of Even and Odd Functions Theorem. A function is even if and only if its graph is symmetric with respect to the y-axis. A function is odd if and only if its graph is symmetric with respect to the origin.

43. Description of Even and Odd Functions Example. Problem: Does the graph represent a function which is even, odd, or neither? Answer:

44. Description of Even and Odd Functions Example. Problem: Does the graph represent a function which is even, odd, or neither? Answer:

45. Description of Even and Odd Functions Example. Problem: Does the graph represent a function which is even, odd, or neither? Answer:

46. Identifying Even and Odd Functions from the Equation Example. Determine whether the following functions are even, odd or neither. (a) Problem: Answer: (b) Problem: g(x) = 3x 2 { 4 Answer: (c) Problem: Answer:

47. Increasing, Decreasing and Constant Functions Increasing function (on an open interval I): For any choice of x 1 and x 2 in I, with x 1 < x 2, we have f(x 1 ) < f(x 2 ) Decreasing function (on an open interval I) For any choice of x 1 and x 2 in I, with x 1 f(x 2 ) Constant function (on an open interval I) For all choices of x in I, the values f(x) are equal.

48. Increasing, Decreasing and Constant Functions

49. Example. Answer the questions about the function shown. (a) Problem: Where is the function increasing? Answer: (b) Problem: Where is the function decreasing? Answer: (c) Problem: Where is the function constant Answer:

50. Increasing, Decreasing and Constant Functions WARNING! Describe the behavior of a graph in terms of its x-values. Answers for these questions should be open intervals.

51. Local Extrema Local maximum at c: Open interval I containing x so that, for all x · c in I, f(x) · f(c). f(c) is a local maximum of f. Local minimum at c: Open interval I containing x so that, for all x · c in I, f(x) ¸ f(c). f(c) is a local minimum of f. Local extrema: Collection of local maxima and minima

52. Local Extrema For local maxima: Graph is increasing to the left of c Graph is decreasing to the right of c. For local minima: Graph is decreasing to the left of c Graph is increasing to the right of c.

53. Local Extrema Example. Answer the questions about the given graph of f. (a) Problem: At which number(s) does f have a local maximum? Answer: (b) Problem: At which number(s) does f have a local minimum? Answer:

54. Average Rate of Change Slope of a line can be interpreted as the average rate of change Average rate of change: If c is in the domain of y = f(x) Also called the difference quotient of f at c

55. Average Rate of Change Example. Find the average rates of change of (a) Problem: From 0 to 1. Answer: (b) Problem: From 0 to 3. Answer: (c) Problem: From 1 to 3: Answer:

56. Secant Lines Geometric interpretation to the average rate of change Label two points (c, f(c)) and (x, f(x)) Draw a line containing the points. This is the secant line. Theorem. [Slope of the Secant Line] The average rate of change of a function equals the slope of the secant line containing two points on its graph

57. Secant Lines

58. Example. Problem: Find an equation of the secant line to containing (0, f(0)) and (5, f(5)) Answer:

59. Key Points Even and Odd Functions Description of Even and Odd Functions Identifying Even and Odd Functions from the Equation Increasing, Decreasing and Constant Functions Local Extrema Average Rate of Change

60. Key Points (cont.) Secant Lines

61. Linear Functions and Models Section 2.4

62. Linear Functions Linear function: Function of the form f(x) = mx + b Graph: Line with slope m and y-intercept b. Theorem. [Average Rate of Change of Linear Function] Linear functions have a constant average rate of change. The constant average rate of change of f(x) = mx + b is

63. Linear Functions Example. Problem: Graph the linear function f(x) = 2x { 5 Answer:

64. Application: Straight-Line Depreciation Example. Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The company chooses to depreciate the machine using the straight-line method over 10 years. For straight-line depreciation, the value of the asset declines by a fixed amount every year.

65. Example. (cont.) (a) Problem: Write a linear function that expresses the book value of the machine as a function of its age, x Answer: (b) Problem: Graph the linear function Answer: Application: Straight-Line Depreciation

66. Example. (cont.) (c) Problem: What is the book value of the machine after 4 years? Answer: (d) Problem: When will the machine be worth $20,000? Answer: Application: Straight-Line Depreciation

67. Scatter Diagrams Example. The amount of money that a lending institution will allow you to borrow mainly depends on the interest rate and your annual income. The following data represent the annual income, I, required by a bank in order to lend L dollars at an interest rate of 7.5% for 30 years.

68. Scatter Diagrams Example. (cont.) Annual Income, I ($) Loan Amount, L ($) 15,00044,600 20,00059,500 25,00074,500 30,00089,400 35,000104,300 40,000119,200 45,000134,100 50,000149,000 55,000163,900 60,000178,800 65,000193,700 70,000208,600

69. Scatter Diagrams Example. (cont.) Problem: Use a graphing utility to draw a scatter diagram of the data. Answer:

70. Linear and Nonlinear Relationships Linear Nonlinear Linear NonlinearLinearNonlinear

71. Line of Best Fit For linearly related scatter diagram Line is line of best fit. Use graphing calculator to find Example. (a) Problem: Use a graphing utility to find the line of best fit to the data in the last example. Answer:

72. Line of Best Fit Example. (cont.) (b) Problem: Graph the line of best fit from the last example on the scatter diagram. Answer:

73. Line of Best Fit Example. (cont.) (c) Problem: Determine the loan amount that an individual would qualify for if her income is $42,000. Answer:

74. Direct Variation Variation or proportionality. y varies directly with x, or is directly proportional to x: There is a nonzero number such that y = kx. k is the constant of proportionality.

75. Direct Variation Example. Suppose y varies directly with x. Suppose as well that y = 15 when x = 3. (a) Problem: Find the constant of proportionality. Answer: (b) Problem: Find x when y = 124.53. Answer:

76. Key Points Linear Functions Application: Straight-Line Depreciation Scatter Diagrams Linear and Nonlinear Relationships Line of Best Fit Direct Variation

77. Library of Functions; Piecewise-defined Functions Section 2.5

78. Linear Functions f(x) = mx+b, m and b a real number Domain: ({ 1, 1 ) Range: ({ 1, 1 ) unless m = 0 Increasing on ({ 1, 1 ) (if m > 0) Decreasing on ({ 1, 1 ) (if m < 0) Constant on ({ 1, 1 ) (if m = 0)

79. Constant Function f(x) = b, b a real number Special linear functions Domain: ({ 1, 1 ) Range: f b g Even/odd/neither: Even (also odd if b = 0) Constant on ({ 1, 1 ) x-intercepts: None (unless b = 0) y-intercept: y = b.

80. Identity Function f(x) = x Special linear function Domain: ({ 1, 1 ) Range: ({ 1, 1 ) Even/odd/neither: Odd Increasing on ({ 1, 1 ) x-intercepts: x = 0 y-intercept: y = 0.

81. Square Function f(x) = x 2 Domain: ({ 1, 1 ) Range: [0, 1 ) Even/odd/neither: Even Increasing on (0, 1 ) Decreasing on ({ 1, 0) x-intercepts: x = 0 y-intercept: y = 0.

82. Cube Function f(x) = x 3 Domain: ({ 1, 1 ) Range: ({ 1, 1 ) Even/odd/neither: Odd Increasing on ({ 1, 1 ) x-intercepts: x = 0 y-intercept: y = 0.

83. Square Root Function Domain: [0, 1 ) Range: [0, 1 ) Even/odd/neither: Neither Increasing on (0, 1 ) x-intercepts: x = 0 y-intercept: y = 0

84. Cube Root Function Domain: ({ 1, 1 ) Range: ({ 1, 1 ) Even/odd/neither: Odd Increasing on ({ 1, 1 ) x-intercepts: x = 0 y-intercept: y = 0

85. Reciprocal Function Domain: x  0 Range: x  0 Even/odd/neither: Odd Decreasing on ({ 1, 0) [ (0, 1 ) x-intercepts: None y-intercept: None

86. Absolute Value Function f(x) = j x j Domain: ({ 1, 1 ) Range: [0, 1 ) Even/odd/neither: Even Increasing on (0, 1 ) Decreasing on ({ 1, 0) x-intercepts: x = 0 y-intercept: y = 0

87. Absolute Value Function Can also write the absolute value function as This is a piecewise-defined function.

88. Greatest Integer Function f(x) = int(x) greatest integer less than or equal to x Domain: ({ 1, 1 ) Range: Integers ( Z ) Even/odd/neither: Neither y-intercept: y = 0 Called a step function

89. Greatest Integer Function

90. Piecewise-defined Functions Example. We can define a function differently on different parts of its domain. (a) Problem: Find f({2) Answer: (b) Problem: Find f({1) Answer: (c) Problem: Find f(2) Answer: (d) Problem: Find f(3) Answer:

91. Key Points Linear Functions Constant Function Identity Function Square Function Cube Function Square Root Function Cube Root Function Reciprocal Function Absolute Value Function

92. Key Points (cont.) Greatest Integer Function Piecewise-defined Functions

93. Graphing Techniques: Transformations Section 2.6

94. Transformations Use basic library of functions and transformations to plot many other functions. Plot graphs that look “ almost ” like one of the basic functions.

95. Shifts Example. Problem: Plot f(x) = x 3, g(x) = x 3 { 1 and h(x) = x 3 + 2 on the same axes Answer:

96. Shifts Vertical shift: A real number k is added to the right side of a function y = f(x), New function y = f(x) + k Graph of new function: Graph of f shifted vertically up k units (if k > 0) Down j k j units (if k < 0)

97. Shifts Example. Problem: Use the graph of f(x) = j x j to obtain the graph of g(x) = j x j + 2 Answer:

98. Shifts Example. Problem: Plot f(x) = x 3, g(x) = (x { 1) 3 and h(x) = (x + 2) 3 on the same axes Answer:

99. Shifts Horizontal shift: Argument x of a function f is replaced by x { h, New function y = f(x { h) Graph of new function: Graph of f shifted horizontally right h units (if h > 0) Left j h j units (if h < 0) Also y = f(x + h) in latter case

100. Shifts Example. Problem: Use the graph of f(x) = j x j to obtain the graph of g(x) = j x+2 j Answer:

101. Shifts Example. Problem: The graph of a function y = f(x) is given. Use it to plot g(x) = f(x { 3) + 2 Answer:

102. Compressions and Stretches Example. Problem: Plot f(x) = x 3, g(x) = 2x 3 and on the same axes Answer:

103. Compressions and Stretches Vertical compression/stretch: Right side of function y = f(x) is multiplied by a positive number a, New function y = af(x) Graph of new function: Multiply each y-coordinate on the graph of y = f(x) by a. Vertically compressed (if 0 < a < 1) Vertically stretched (if a > 1)

104. Compressions and Stretches Example. Problem: Use the graph of f(x) = x 2 to obtain the graph of g(x) = 3x 2 Answer:

105. Compressions and Stretches Example. Problem: Plot f(x) = x 3, g(x) = (2x) 3 and on the same axes Answer:

106. Compressions and Stretches Horizontal compression/stretch: Argument x of a function y = f(x) is multiplied by a positive number a New function y = f(ax) Graph of new function: Divide each x-coordinate on the graph of y = f(x) by a. Horizontally compressed (if a > 1) Horizontally stretched (if 0 < a < 1)

107. Compressions and Stretches Example. Problem: Use the graph of f(x) = x 2 to obtain the graph of g(x) = (3x) 2 Answer:

108. Compressions and Stretches Example. Problem: The graph of a function y = f(x) is given. Use it to plot g(x) = 3f(2x) Answer:

109. Reflections Example. Problem: f(x) = x 3 + 1 and g(x) = {(x 3 + 1) on the same axes Answer:

110. Reflections Reflections about x-axis : Right side of the function y = f(x) is multiplied by {1, New function y = {f(x) Graph of new function: Reflection about the x-axis of the graph of the function y = f(x).

111. Reflections Example. Problem: f(x) = x 3 + 1 and g(x) = ({x) 3 + 1 on the same axes Answer:

112. Reflections Reflections about y-axis : Argument of the function y = f(x) is multiplied by {1, New function y = f({x) Graph of new function: Reflection about the y-axis of the graph of the function y = f(x).

113. Summary of Transformations

116. Example. Problem: Use transformations to graph the function Answer:

117. Key Points Transformations Shifts Compressions and Stretches Reflections Summary of Transformations

118. Mathematical Models: Constructing Functions Section 2.7

119. Mathematical Models Example. Problem: The volume V of a right circular cylinder is V = ¼r 2 h. If the height is three times the radius, express the volume V as a function of r. Answer:

120. Mathematical Models Example. Anne has 5000 feet of fencing available to enclose a rectangular field. One side of the field lies along a river, so only three sides require fencing. (a) Problem: Express the area A of the rectangle as a function of x, where x is the length of the side parallel to the river. Answer:

121. Mathematical Models Example (cont.) (b) Problem: Graph A = A(x) and find what value of x makes the area largest. Answer: (c) Problem: What value of x makes the area largest? Answer:

122. Key Points Mathematical Models